Turns out it might be due to New England provincialism.
Spring and fall likely gained popularity in conjunction with each other. They each initially appeared in the 16th century as spring of the leaf and fall of the leaf, respectively. The two complemented each other nicely and were soon shortened to the more succinct fall and spring, with the longer phrases disappearing over the next few hundred years.
It’s a bit of a mystery why the superfluous autumn persists while analogous words like primetemps and ver have fallen out of use, but it may have something to do with the Atlantic Ocean. The rise of autumn and the appearance of fall happened around the same time as the British arrival on the American continent, and it’s there that the latter really caught on.